I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material covered was the bare bone basics of derived functors as universal $\delta$-functors, projective, injective, and acyclic resolutions, and nice applications to algebraic geometry, group cohomology, and sheaf cohomology.

I want to show the relevance of homotopical algebra to homological algebra. However, the lecturer is not too interested in abstract nonsense, so my original idea of showing the general definition of derived functors as 'homotopical approximations' in homotopical categories is not, by itself, a very promising option.

What I'm looking for are simple "applications" of homotopical ideas to homological algebra, which are elementary in the sense of not requiring too many deep results in homotopy theory and algebraic topology (categorical machinery is not constrained though). Things such as spectra, stable/unstable homotopy theory, etc are to be avoided. By applications I do not necessarily mean explicit computations, by the way.

Understandably, people may be tempted to say the homological algebra is just a special case of homotopical algebra, e.g as in Adeel Khan's comment on this question, and also delve into the realm of $\infty$-categories/topoi, but these ideas, which I feel are huge conceptual advances, are far beyond my level and time limit. I'm really looking for "basic" stuff.

Regretably, my question is ill-posed, but I feel this is inevitable.

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    $\begingroup$ I think wanting to present homological algebra as a special case of homotopical algebra would be somewhat putting the cart before the horses. It's my understanding that homotopical algebra was more-or-less developed as a far-reaching generalization of homological algebra. Just like you probably wouldn't make a presentation on manifolds for a first course on multivariable calculus, I don't think it would be a great idea (but I'm open to being corrected) to make a presentation on homotopical algebra after just learning about the basics of homological algebra... $\endgroup$ – Najib Idrissi Jun 5 '15 at 13:09
  • $\begingroup$ You could say something on the nonabelian tensor product - see the bibliography at pages.bangor.ac.uk/~mas010/nonabtens.html. The general definition came from homotopical considerations, see [6] on the list, but there are relations with homological algebra, see [1], and there are fun calculations, as shown by the list. $\endgroup$ – Ronnie Brown Jun 5 '15 at 13:41
  • $\begingroup$ @NajibIdrissi you make a fair point. However, it is not really my aim to present homotopy theory. Initially, I just wanted to present derived functors as homotopical approximations for reasons of elegance and conceptual clarity. To me $\delta$-functors feel awkward, and upon seeing comments like the one mentioned in the question, it felt being homotopical is the "right" property - as opposed to splicing long exact sequences. In response to your manifold analogy, I feel I would like to emphasize the derivative as a linear operator at the end of a first course. $\endgroup$ – Arrow Jun 8 '15 at 17:11

The application that convinced me I should care about homotopical ideas is an answer to the following question:

Where do long exact sequences come from?

My answer is at this MO question.


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